arima

Syntax

Description

ARIMAX = arima(Mdl) converts
the univariate regression model with ARIMA time series errors Mdl to
a model of type arima including
a regression component (ARIMAX).

[ARIMAX,XNew]
= arima(Mdl,Name,Value) returns
an updated regression matrix of predictor data using additional options
specified by one or more Name,Value pair arguments.

Input Arguments

Mdl

Regression model with ARIMA time series errors, as created by regARIMA or estimate.

Name-Value Pair Arguments

Specify optional
comma-separated pairs of Name,Value arguments. Name is
the argument name and Value is the corresponding value.
Name must appear inside single quotes (' '). You can
specify several name and value pair arguments in any order as
Name1,Value1,...,NameN,ValueN.

'X'

Predictor data for the regression component of Mdl,
specified as the comma-separated pair consisting of 'X' and
a matrix.

The last row of X contains the latest observation
of each series.

Each column of X is a separate time series.

Output Arguments

ARIMAX

ARIMAX model equivalent to the regression model with ARIMA errors Mdl,
returned as a model of type arima.

XNew

Updated predictor data matrix for the regression component of ARIMAX,
returned as a matrix.

XNew has the same number of rows as X.
The last row of XNew contains the latest observation
of each series.

Each column of XNew is a separate time series.
The number of columns of XNew is one plus the number
of nonzero autoregressive coefficients in the difference equation
of Mdl.

and is row j of X. Since the product of the autoregressive and integration polynomials is ARIMAX.Beta is simply [1 -0.8 0.4]. Note that the software carries over the autoregressive and moving average coefficients from Mdl to ARIMAX. Also, Mdl.Intercept = 1 and ARIMAX.Constant = (1 - 0.8 + 0.4)(1) = 0.6, i.e., the regARIMA model intercept and arima model constant are generally unequal.

Mdl.Beta has length 2, but ARIMAX.Beta has length 8. This is because the product of the autoregressive and integration polynomials, , is

You can see that when you add seasonality, seasonal lag terms, and integration to a model, the size of XNew can grow quite large. A conversion such as this might not be ideal for analyses involving small sample sizes.

Algorithms

If you specify X, then arima returns XNew in
a certain format. Suppose that the nonzero autoregressive lag term
degrees of Mdl are 0 < a1 < a2 <
...< P, which is the largest lag term degree.
The software obtains these lag term degrees by expanding and reducing
the product of the seasonal and nonseasonal autoregressive lag polynomials,
and the seasonal and nonseasonal integration lag polynomials

ϕ(L)(1−L)DΦ(L)(1−Ls).

The first column of XNew is Xβ.

The second column of XNew is a
sequence of a1NaNs,
and then the product Xa1β, where Xa1β=La1Xβ.

The jth column of XNew is
a sequence of ajNaNs,
and then the product Xajβ, where Xajβ=LajXβ.

The last column of XNew is a sequence
of apNaNs,
and then the product Xpβ, where Xpβ=LpXβ.

Suppose that Mdl is a regression
model with ARIMA(3,1,0) errors, and ϕ1 =
0.2 and ϕ3 = 0.05.
Then the product of the autoregressive and integration lag polynomials
is